A. Technical details for the proof of (Aletti, Crimaldi and Ghiglietti, 2019, Lemma 5.1)

Supplementary Material for

“Interacting Reinforced Stochastic Processes:

Statistical Inference based on the Weighted Empirical Means”

GIACOMO ALETTI^1,∗ IRENE CRIMALDI^2,† and ANDREA GHIGLIETTI³

1ADAMSS Center, Universit`a degli Studi di Milano, Milan, Italy E-mail:giacomo.aletti@unimi.it

2IMT School for Advanced Studies, Lucca, Italy E-mail:irene.crimaldi@imtlucca.it

3LivaNova, Milan, Italy

E-mail:Andrea.Ghiglietti@guest.unimi.it

In this document we collect some technical results and computations necessary for the proof of (Aletti, Crimaldi and Ghiglietti,2019, Lemma5.1) and we give the entries of the matrices S_γ¹¹, S_γ¹²and S_γ²²introduced in (Aletti, Crimaldi and Ghiglietti, 2019, Theorem5.1). Therefore, the notation and the assumptions used here are the same as those used in that paper.

A. Technical details for the proof of (Aletti, Crimaldi and Ghiglietti, 2019, Lemma 5.1)

In all the sequel, given (zn)n, (z_n⁰)n two sequences of complex numbers, the notation zn = O(z_n⁰) means

|zn| ≤ C|z⁰_n| for a suitable constant C > 0 and n large enough. Moreover, if z_n⁰ 6= 0, the notation zn∼ zz_n⁰ with z ∈ C \ {0} means limⁿzn/z_n⁰ = z and, finally, the notation zn= o(z⁰_n) means limnzn/z_n⁰ = 0.

Given 1/2 < δ ≤ 1, x = ax+ i bx∈ C with ax > 0 and an integer m0 ≥ 2 such that axm^−δ < 1 for all m ≥ m0, let us set

p^δ_n(x) :=

n

Y

m=m0

 1 − x

m^δ



for n ≥ m0. (A.1)

A.1. Some technical results

We first recall the following result, which has been proved inAletti, Crimaldi and Ghiglietti(2017).

Lemma A.1. (Aletti, Crimaldi and Ghiglietti,2017, Lemma A.4) We have

|p^δ_n(x)| = (O

exp

−a_xⁿ_1−δ^1−δ

for 1/2 < δ < 1 O (n^−ax) for δ = 1

(A.2)

∗Member of “Gruppo Nazionale per il Calcolo Scientifico (GNCS)” of the Italian Institute “Istituto Nazionale di Alta Matematica (INdAM)”

†Member of “Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA)” of the Italian Institute “Istituto Nazionale di Alta Matematica (INdAM)”

1

and

|p^δ_n(x)⁻¹| = (O

exp axn^1−δ

1−δ



for 1/2 < δ < 1 O (n^ax) for δ = 1 .

(A.3)

Therefore , if we set

F_k+1,n^δ (x) := p^δ_n(x)

p^δ_k(x) for m0≤ k ≤ n , (A.4)

we have

|F_k+1,n^δ (x)| =





 O

exp

ax

1−δ(k^1−δ− n^1−δ)

for 1/2 < δ < 1 O

k n

^ax

for δ = 1.

(A.5)

Now, we prove two other results.

Lemma A.2. Given β > 1 and e > 0, we have

n

X

k=m₀

1

k^β|F_k+1,n^δ (x)|^e =











O n^−(β−δ)

if 1/2 < δ < 1,

O (n^−eax) if δ = 1 and eax< β − 1, O n^−(β−1)ln(n)

if δ = 1 and eax= β − 1, O n^−(β−1)

if δ = 1 and eax> β − 1.

(A.6)

Proof. The desired relations immediately follows from (A.5) using the well-known relation

n

X

k=1

1 k^1−a =











O(1) for a < 0,

ln(n) + d + O(n⁻¹) = ln(n) + O(1) for a = 0, a⁻¹n^a+ O(1) for 0 < a ≤ 1,

a⁻¹n^a+ O(n^a−1) for a > 1,

(A.7)

where d is the Euler-Mascheroni constant, and the relation

n

X

k=1

exp(ak^b/b)

k^β = O

Z n 1

exp(at^b/b) t^β dt



= O exp(at^b/b) at^b+β−1

ⁿ

1

+(b + β − 1) a

Z n 1

exp(at^b/b) t^β+b dt



= O exp(an^b/b) n^b+β−1



for a > 0, b > 0, β > 1.

(A.8)

Indeed, for the case δ = 1, it is enough to apply (A.7) with a = ea_x− (β − 1); while, for the case 1/2 < δ < 1, it is enough to apply (A.8) with a = ea_x, b = 1 − δ and β.

The following lemma extends (Aletti, Crimaldi and Ghiglietti,2017, Lemma A.5).

Lemma A.3. Given 1/2 < δ1≤ δ2≤ 1, β > δ1 and x1, x2∈ C with Re(x1) > 0, Re(x2) > 0, let m0≥ 2 be an integer such that max{Re(x1), Re(x2)}m^−δ1 < 1 for all m ≥ m0. Then we have

limn n^β−δ1

n

X

k=m0

k^−βF_k+1,n^δ1 (x1)F_k+1,n^δ2 (x2) =











1

x₁+x₂ if 1/2 < δ₁= δ₂< 1,

1

x₁+x₂−β+1 if δ₁= δ₂= 1 and Re(x₁+ x₂) > β − 1,

1

x₁ if 1/2 < δ₁< δ₂≤ 1.

(A.9)

Proof. Let us start with observing that, in each considered case, relation (A.2) implies

limn n^β−δ1|p^δ_n¹(x1)| |p^δ_n²(x2)| = 0. (A.10) Indeed, in particular, when δ₁= δ₂= 1 we have the additional condition Re(x₁+ x₂) > β − 1.

Now, fix k ≥ 2 and let us set η := β − δ₁and `^δ_n(x) := 1/p^δ_n(x) and define the following quantity Dk := 1

k^η`<sup>δ</sup><sub>k</sub><sup>1</sup>(x1)`^δ_k²(x2) − 1

(k − 1)^η`<sup>δ</sup><sub>k−1</sub><sup>1</sup> (x1)`^δ_k−1² (x2)

=  1

k^η − 1 (k − 1)^η



`<sup>δ</sup><sub>k−1</sub><sup>1</sup> (x<sub>1</sub>)`^δ_k−1² (x₂) + 1 k^η

`<sup>δ</sup><sub>k</sub><sup>1</sup>(x<sub>1</sub>)`^δ_k²(x₂) − `<sup>δ</sup><sub>k−1</sub><sup>1</sup> (x<sub>1</sub>)`^δ_k−1² (x₂)

= `<sup>δ</sup><sub>k</sub><sup>1</sup>(x1)`^δ_k²(x2)

"

 1

k^η − 1 (k − 1)^η

`<sup>δ</sup><sub>k−1</sub><sup>1</sup> (x<sub>1</sub>)`^δ_k−1² (x₂)

`<sup>δ</sup><sub>k</sub><sup>1</sup>(x1)`^δ_k²(x2) + 1

k^η 1 − `<sup>δ</sup><sub>k−1</sub><sup>1</sup> (x<sub>1</sub>)`^δ_k−1² (x₂)

`<sup>δ</sup><sub>k</sub><sup>1</sup>(x1)`^δ_k²(x2)

!#

. Then, we observe the following:

 1

k^η − 1 (k − 1)^η



= − η k^1+η + O

 1 k^2+η



for k → +∞ (A.11)

and

`<sup>δ</sup><sub>k−1</sub><sup>1</sup> (x1)`^δ_k−1² (x2)

`<sup>δ</sup><sub>k</sub><sup>1</sup>(x<sub>1</sub>)`^δ_k²(x₂) =  1 − x1

k^δ1

  1 − x2

k^δ2



= 1 + x1x2

k^(δ1+δ2)− x1

k^δ1 − x2

k^δ2. (A.12) Now, by using (A.11) and (A.12) in the above expression of D_k, we have for k → +∞

Dk = `<sup>δ</sup><sub>k</sub><sup>1</sup>(x1)`^δ_k²(x2)



− η

k^η+1 + O(1/k^η+2) 

1 + x1x2

k^(δ1+δ2) − x1

k^δ1 − x2

k^δ2

 + 1

k^η

− x1x2

k^(δ1+δ2) + x1

k^δ1 + x2

k^δ2



=

(`<sup>δ</sup><sub>k</sub>(x1)`_k^δ(x2)x1+x2

k^η+δ −_kη+1^η + o(1/k^η+δ)

if δ1= δ2= δ,

`<sup>δ</sup><sub>k</sub><sup>1</sup>(x<sub>1</sub>)`^δ_k²(x₂) x₁

kη+δ1 + o(1/k^η+δ1)

if δ₁< δ₂

=







`<sup>δ</sup><sub>k</sub>(x1)`^δ_k(x2)^x_k¹η+δ^+x2 + o(`<sup>δ</sup><sub>k</sub>(x1)`^δ_k(x2)/k^η+δ) if δ1= δ2= δ < 1,

`<sup>1</sup><sub>k</sub>(x<sub>1</sub>)`¹_k(x₂)^x1+x_kη+1^2−η + o(`<sub>k</sub><sup>1</sup>(x<sub>1</sub>)`¹_k(x₂)/k^η+1) if δ₁= δ₂= 1 and Re(x₁+ x₂) > η,

`<sub>k</sub><sup>δ1</sup>(x1)`^δ_k²(x2)_kη+δ1^x1 + o(`<sub>k</sub><sup>δ1</sup>(x1)`^δ_k²(x2)/k^η+δ1) if δ1< δ2. that is

D_k ∼







x₁+x₂

k^η+δ `<sup>δ</sup><sub>k</sub>(x<sub>1</sub>)`_k^δ(x₂) if 1/2 < δ₁= δ₂= δ < 1,

x₁+x₂−η

k^η+1 `<sup>1</sup><sub>k</sub>(x1)`¹_k(x2) if δ1= δ2= 1 and Re(x1+ x2) > η,

x₁

kη+δ1 `<sup>δ</sup><sub>k</sub><sup>1</sup>(x<sub>1</sub>)`_k^δ2(x₂) if 1/2 < δ₁< δ₂≤ 1.

(A.13)

Now, following the same arguments used in the proof of (Aletti, Crimaldi and Ghiglietti,2017, Lemma A.5), in order to conclude, we apply (Aletti, Crimaldi and Ghiglietti,2017, Corollary A.2) with

z_n = D_n, v_n= n^ηp^δ_n¹(x₁)p^δ_n²(x₂), w_n= `<sub>n,1</sub>`_n,2 n^η+δ1Dn

, w =







1

x₁+x₂ if 1/2 < δ1= δ2= δ < 1

1

x1+x2−η if δ1= δ2= 1 and Re(x1+ x2) > η

1

x₁ if 1/2 < δ1< δ2≤ 1.

Indeed, limnvn= 0 by (A.10), limnwn = w 6= 0 by (A.13), limn v_n

n

X

k=m₀

z_k= lim

n n^ηp^δ_n¹(x₁)p^δ_n²(x₂)

n

X

k=m₀

D_k

= lim

n n^ηp^δ_n¹(x₁)p^δ₂²(x₂) `<sup>δ</sup><sub>n</sub><sup>1</sup>(x<sub>1</sub>)`^δ_n²(x₂) n^η −`^δ_m¹

0−1(x1)`^δ_m²

0−1(x2) (m0− 1)^η

!

= 1 by (A.10) and z_n⁰ = z_nw_n= r²_n`<sub>n,1</sub>`_n,2.

A.2. Analytic expression of A

with j ≥ 2

Let us recall the definition of the following quantities for j ≥ 2:

A^j_k+1,n−1=

n−1

Y

m=k+1

(I − DQ,j,m) , where DQ,j,n=



brn−1(1 − λj) 0

−λjhn(λj) qbn,n



with the function h_n defined in (Aletti, Crimaldi and Ghiglietti, 2019, Equation (5.10)). The aim of this section is to compute the product above and so finding the following useful expression of A^j_k+1,n−1:

A^j_k+1,n−1=

 F_k+1,n−1^γ (cαj) 0 λjGk+1,n−1(cαj, q) F_k+1,n−1^ν (q)

 , where

Gk+1,n−1(cαj, q) =

n−1

X

l=k+1

F_l+1,n−1^γ (cαj)hl(λj)F_k+1,l−1^ν (q).

It is straightforward to see that [A^j_k+1,n−1]₂₁= 0,

[A^j_k+1,n−1]11 =

n−1

Y

m=k+1

(1 −rbm−1(1 − λj)) = F_k+1,n−1^γ (cαj),

[A^j_k+1,n−1]22 =

n−1

Y

m=k+1

(1 −bqm,m) = F_k+1,n−1^ν (q),

while it is not immediate to determine [A^j_k+1,n−1]₁₂. To this end, let us set x_n−1:= [A^j_k+1,n−1]₂₁and observe that, since A^j_k+1,n−1= A^j_k+1,n−2(I − DQ,j,n−1) and xk+1= λjhk+1(λj), we have that

xn−1= xn−2(1 −brn−1(1 − λj)) + [A^j_k+1,n−2]22λjhn−1(λj)

= xn−2F_n−1,n−1^γ (cαj) + F_k+1,n−2^ν (q)λjhn−1(λj)

= x_n−3F_n−2,n−1^γ (cα_j) + F_k+1,n−3^ν (q)λ_jh_n−2(λ_j)F_n−1,n−1^γ (cα_j) + F_k+1,n−2^ν (q)λ_jh_n−1(λ_j)

= . . .

= x_k+1F_k+2,n−1^γ (cα_j) +

n−2

X

l=k+2

F_k+1,l−1^ν (q)λ_jh_l(λ_j)F_l+1,n−1^γ (cα_j) + F_k+1,n−2^ν (q)λ_jh_n−1(λ_j)

=

n−1

X

l=k+1

F_k+1,l−1^ν (q)λjhl(λj)F_l+1,n−1^γ (cαj)

= λjGk+1,n−1(cαj, q) .

A.3. Asymptotic behavior of G

(x, q)

Let us recall the definition

Gk+1,n−1(x, q) :=

n−1

X

l=k+1

F_l+1,n−1^γ (x)hl(1 − c⁻¹x)F_k+1,l−1^ν (q).

Here we prove the following result:

Lemma A.4. When ν = γ, we have for x ∈ C \ {0}

Gk+1,n−1(x, q) =











q x−q

F_k+1,n−1^γ (q) − F_k+1,n−1^γ (x)

if x 6= q,

q

1−γF_k+1,n−1^γ (q)(n − 1)^1−γ− (k + 1)^1−γ + O(F_k+1,n−1^γ (q)) if x = q and 1/2 < γ < 1, qF_k+1,n−1^γ (q) ln

n−1 k+1



+ O(k⁻¹F_k+1,n−1^γ (q)) if x = q and γ = 1.

(A.14) When ν 6= γ, we have for x ∈ C \ {0}

G_k+1,n−1(x, q) = C(x, q) F_k+1,n−1^ν (q)

(n − 1)^µ −F_k+1,n−1^γ (x) k^µ

!

+ O |F_k+1,n−1^ν (q)|

n^2µ +|F_k+1,n−1^γ (x)|

k^2µ

!

, (A.15)

where µ := |γ − ν| and

C(x, q) :=

(−^x_q if ν < γ,

q

x if γ < ν.

Proof. Recalling the definition (A.4), we can write

G_k+1,n−1(x, q) =

n−1

X

l=k+1

F_l+1,n−1^γ (x)h_l(1 − c⁻¹x)F_k+1,l−1^ν (q) =

n−1

X

l=k+1

p^γ_n−1(x)

p^γ_l(x) h_l(1 − c⁻¹x)p^ν_l−1(q) p^ν_k(q)

= p^γ_n−1(x) p^ν_k(q)

n−1

X

l=k+1

hl(1 − c⁻¹x)

(1 −qb_l,l) Xl, where Xl := p^ν_l(q) p^γ_l(x).

(A.16)

Moreover, recalling the definition of the function h_l (see (Aletti, Crimaldi and Ghiglietti, 2019, Equation (5.10))), we have for x 6= 0

hl(1 − c⁻¹x) = (

br_l−1c⁻¹x = xl^−γ if ν < γ, bql,l= ql^−ν if ν ≥ γ.

Let us start with the case ν = γ. In this case, we have

∆Xl:= Xl− Xl−1=



1 −Xl−1

X_l



Xl= x − q q

q l^γ(1 − ql^−γ)

 Xl

= x − q q

qb_l,l 1 −qbl,l

Xl=x − q q

h_l(1 − c⁻¹x) 1 −qbl,l

Xl. It follows that

x − q q

n−1

X

l=k+1

h_l(1 − c⁻¹x) 1 −qbl,l

X_l= X_n−1− Xk. Since

p^γ_n−1(x)

p^ν_k(q) Xn−1 = p^γ_n−1(x) p^ν_k(q)

p^ν_n−1(q)

p^γ_n−1(x) = F_k+1,n−1^ν (q) and p^γ_n−1(x)

p^ν_k(q) X_k = p^γ_n−1(x) p^ν_k(q)

p^ν_k(q)

p^γ_k(x) = F_k+1,n−1^γ (x),

(A.17)

we find by (A.16)

x − q

q Gk+1,n−1(x, q) =

F_k+1,n−1^γ (q) − F_k+1,n−1^γ (x)

and so for x 6= q we get

Gk+1,n−1(x, q) = q x − q



F_k+1,n−1^γ (q) − F_k+1,n−1^γ (x) .

When ν = γ and x = q, we have X_l= 1 and so we obtain (by (A.16) together with (A.7))

G_k+1,n−1(x, q) = qF_k+1,n−1^γ (q)

n−1

X

l=k+1

1

l^γ(1 − ql^−γ) = qF_k+1,n−1^γ (q)

n−1

X

l=k+1

1 l^γ + O



 X

l≥k+1

l^−2γ





= qF_k+1,n−1^γ (q)

(_(n−1)1−γ

1−γ −^(k+1)_1−γ^1−γ + O(1) + O(k^−(2γ−1)) if 1/2 < γ < 1 ln(n − 1) − ln(k + 1) + O(n⁻¹) + O(k⁻¹) if γ = 1, which implies the two different asymptotic behavior in (A.14) according to the value of γ.

Now, let us consider the case ν 6= γ and introduce the sequence {y_l; l ≥ 1} defined as y_l := l^−µ, with µ = |γ − ν|. Then, we have

ylXl− yl−1Xl−1= ∆ylXl+ yl−1∆Xl= 1

l^µ − 1 (l − 1)^µ



Xl+ 1 l^µ + O

 1 l^1+µ



∆Xl

= −µ l^1+µ + O

 1 l^2+µ



Xl+ 1 l^µ + O

 1 l^1+µ



∆Xl, where

∆Xl:= Xl− Xl−1=



1 −Xl−1

X_l



Xl= RlXl

with

Rl:=



1 −Xl−1

X_l



= xl^−γ− ql^−ν

1 − ql^−ν = brl−1c⁻¹x −bql,l

1 −bq_l,l = O

 1

l^min{γ,ν}

 . Taking into account that µ + min{γ, ν} < 1 + µ for ν 6= γ, we obtain that

ylXl− yl−1Xl−1= R_l l^µ + O

 1 l^1+µ



Xl= K(x, q)h_l(1 − c⁻¹x)

(1 −qbl,l) Xl+ QlXl, where

K(x, q) := 

−q

x 1{ν<γ}+ x q



1{ν>γ}= C(x, q)⁻¹ and

Ql:=

(_xl^{−(2γ−ν)}

1−bq_l,l + O(l^−(1+µ)) if ν < γ,

−^{ql−(2ν−γ)}₁₋

bq_l,l + O(l^−(1+µ)) if ν > γ.

Note that Ql∼ κl−(2µ+min{γ,ν}) with a suitable κ 6= 0. The above expression implies that Xn−1

(n − 1)^µ −Xk

k^µ =

n−1

X

l=k+1

(ylXl− yl−1Xl−1) = K(x, q)

n−1

X

l=k+1

hl(1 − c⁻¹x) (1 −qbl,l) Xl+

n−1

X

l=k+1

QlXl. (A.18)

With similar computations, setting

R^∗_l := 1 − |Xl−1|

|Xl| = |1 − ql^−ν| − |1 − xl^−γ|

|1 − ql^−ν|

and taking into account that R^∗_ll^−2µ∼ κ⁰l−(2µ+min{γ,ν})with a suitable κ⁰ 6= 0 and min{γ, ν} < 1 for ν 6= γ, we find

|Xl|

l^2µ − |Xl−1|

(l − 1)^2µ = R^∗_l l^2µ + O

 1 l^1+2µ



|X_l| = Q^∗_l|X_l|.

Then, since Q_l∼ κ⁰⁰Q^∗_l with a suitable κ⁰⁰6= 0,

n−1

X

l=k+1

QlXl= O

n−1

X

l=k+1

Q^∗_l|Xl|

!

= O

 |Xn−1|

(n − 1)^2µ −|Xk| k^2µ

 . Finally, by (A.16), (A.17), (A.18) and the last above relations, we obtain for x 6= 0

Gk+1,n−1(x, q) = C(x, q) F_k+1,n−1^ν (q)

(n − 1)^µ −F_k+1,n−1^γ (x) k^µ

!

+ O |F_k+1,n−1^ν (q)|

n^2µ +|F_k+1,n−1^γ (x)|

k^2µ

! .

A.4. Asymptotic behavior of C

P

j∈J

Y

Recalling (Aletti, Crimaldi and Ghiglietti, 2019, Equation (5.37)) and taking into account the fact that in all the considered cases with 1 ∈ J , i.e. (ii), (iii) and (v), we have 1 /∈ I1, we get

  C_m^{J (I)}_0,n

 = O(C_n¹¹) + O(C_n²¹) + O(C_n²²), where

C_n¹¹ := X

j∈J, j6=1

1{1∈Ij}|F_m^γ

0,n−1(cαj)| , C_n²¹ := X

j∈J, j6=1

1{2∈I_j}|G_m0,n−1(cα_j, q)| ,

C_n²² := X

j∈J

1{2∈Ij}|F_m^ν_0,n−1(q)| .

Using (A.5) and denoting by a^∗ the real part of α^∗:= 1 − λ^∗, it is immediate to see that

C_n¹¹= X

j∈J, j6=1

1{1∈Ij}





 O

 exp



−ca^∗n^1−γ 1 − γ



if 1/2 < γ < 1

O(n^−ca∗) if γ = 1

and

C_n²²=X

j∈J

1{2∈Ij}





 O

 exp



−qn^1−ν 1 − ν



if 1/2 < ν < 1

O(n^−q) if ν = 1.

For the term C_n²¹, we apply LemmaA.4so that we get:

Case ν < γ We have Gm₀,n−1(cαj, q) = O n^−(γ−ν)|F_m^ν_0,n−1(q)| + |F_m^γ_0,n−1(cαj)|
by means of LemmaA.4 and so

C_n²¹= X

j∈J, j6=1

1{2∈Ij}O

n^−(γ−ν)|F_m^ν_0,n−1(q)| + |F_m^γ_0,n−1(cαj)| ,

where, as above, by (A.5), we have |F_m^ν_0,n−1(q)| = O exp

−qⁿ_1−ν^1−ν

and

|F_m^γ

0,n−1(cα_j)| =





 O

 exp



−ca^∗n^1−γ 1 − γ



if 1/2 < γ < 1

O(n^−ca∗) if γ = 1.

Case ν > γ We have Gm₀,n−1(cαj, q) = O n^−(ν−γ)|F_m^ν_0,n−1(q)| + |F_m^γ

0,n−1(cαj)|
by means of LemmaA.4 and so

C_n²¹= X

j∈J, j6=1

1{2∈Ij}O

n^−(ν−γ)|F_m^ν_0,n−1(q)| + |F_m^γ

0,n−1(cαj)| ,

where, as above, by (A.5), we have |F_m^γ

0,n−1(cαj)| = O exp

−ca^{∗ n}_1−γ^1−γ

and

|F_m^ν_0,n−1(q)| =





 O

 exp



−qn^1−ν 1 − ν



if 1/2 < ν < 1

O(n^−q) if ν = 1.

Case ν = γ Assuming q 6= cα_j for all j ≥ 2, by LemmaA.4, we have¹ G_m0,n−1(cα_j, q) = O |F_m^γ

0,n−1(q)| +

|F_m^γ_0,n−1(cα_j)|
and so

C_n²¹= X

j∈J, j6=1

1{2∈Ij}O |F_m^γ_0,n−1(q)| + |F_m^γ_0,n−1(cαj)|
,

where, as above, by (A.5), we have for x = q or x ∈ {cαj : j ∈ J, j 6= 1}

|F_m^γ_0,n−1(x)| =





 O

 exp



−ax

n^1−γ 1 − γ



if 1/2 < ν = γ < 1

O(n^−ax) if ν = γ = 1

and so, setting x^∗:= min{q, ca^∗}, we can write

C_n²¹= X

j∈J, j6=1

1{2∈Ij}





 O

 exp



−x^∗n^1−γ 1 − γ



if 1/2 < ν = γ < 1

O(n^−x∗) if ν = γ = 1.

Summing up, taking into account the conditions ca^∗ > 1/2 when γ = 1 and q > 1/2 when ν = 1, we can conclude that in all the six cases (i)-(vi) we have tn(J (I))

C_m^{J (I)}_0,n−1  

→ 0 and so tn(J (I))C_m^{J (I)}_0,n−1X

j∈J

Yj,m₀

−→ 0.a.s.

A.5. Asymptotic behavior of P

k=m0

ρ

Recall (Aletti, Crimaldi and Ghiglietti,2019, Equation (5.39)) and the fact that ∆RY,k+1= (∆RZ,k+1, ∆RN,k+1)^>, where, by (Aletti, Crimaldi and Ghiglietti, 2019, Assumption 2.2), we have |∆RZ,k+1| = O(k^−2γ) and

|∆RN,k+1| = O(k^−2ν). Then, taking into account the fact that in all the considered cases with 1 ∈ J , i.e. (ii), (iii) and (v), we have 1 /∈ I1, we get

n−1

X

k=m₀

ρ^{J (I)}_k+1,n−1     

= O(ρ¹¹_n ) + O(ρ²¹_n ) + O(ρ²²_n ),

1If there exists j ≥ 2 such that q = cαj, we have to consider the other asymptotic expression given in LemmaA.4.

where

ρ¹¹_n := X

j∈J, j6=1

1{1∈Ij} n−1

X

k=m0

k^−2γ|F_k+1,n−1^γ (cα_j)| ,

ρ²¹_n := X

j∈J, j6=1

1{2∈Ij} n−1

X

k=m₀

k^−2γ|Gk+1,n−1(cαj, q)| ,

ρ²²_n :=X

j∈J

1{2∈I_j} n−1

X

k=m₀

(k^−2γ+ k^−2ν)|F_k+1,n−1^ν (q)|.

Using Lemma A.2(with β = 2γ > 1, e = 1 and δ = γ), we get

ρ¹¹_n = X

j∈J, j6=1

1{1∈I_j}











O (n^−γ) if 1/2 < γ < 1, O n^−ca∗

if γ = 1 and 1/2 < ca^∗< 1, O n⁻¹ln(n)

if γ = 1 and ca^∗= 1, O n⁻¹

if γ = 1 and ca^∗> 1.

For ρ²²_n , we observe that we have k^−2γ = O(k^−2ν) when ν ≤ γ and k^−2ν = O(k^−2γ) when ν > γ. Therefore, using LemmaA.2 (with e = 1 and δ = ν and β = 2ν > 1 if ν ≤ γ and β = 2γ > 1 if ν > γ), we obtain for the case ν ≤ γ

ρ²²_n =X

j∈J

1{2∈I_j}O

n−1

X

k=m₀

k^−2ν|F_k+1,n−1^ν (q)|

!

=X

j∈J

1{2∈Ij}











O (n^−ν) if 1/2 < ν < 1,

O (n^−q) if ν = 1 and 1/2 < q < 1, O n⁻¹ln(n)

if ν = 1 and q = 1, O n⁻¹

if ν = 1 and q > 1 , and for the case ν > γ

ρ²²_n =X

j∈J

1{2∈Ij}O

n−1

X

k=m0

k^−2γ|F_k+1,n−1^ν (q)|

!

=X

j∈J

1{2∈Ij}











O n^−2γ+ν

if 1/2 < ν < 1,

O (n^−q) if ν = 1 and 1/2 < q < 2γ − 1, O (n^−qln(n)) if ν = 1 and q = 2γ − 1 > 1/2, O n^−2γ+1

if ν = 1 and q > max{1/2, 2γ − 1} .

(A.19)

For the term ρ²¹_n , we apply LemmaA.2and Lemma A.4so that we get:

Case ν < γ We have Gk+1,n−1(cαj, q) = O n^−(γ−ν)|F_k+1,n−1^ν (q)| + k^−(γ−ν)|F_k+1,n−1^γ (cαj)|
by means of LemmaA.4, and so we get

ρ²¹_n = X

j∈J j6=1

1{2∈Ij}O n^−(γ−ν)

n−1

X

k=m₀

1

k^2γ|F_k+1,n−1^ν (q)| +

n−1

X

k=m₀

1

k^3γ−ν|F_k+1,n−1^γ (cαj)|

! ,

where, by LemmaA.2, the first term is O(n^−3γ+2ν), while for the second term we have

n−1

X

k=m₀

1

k^3γ−ν|F_k+1,n−1^γ (cαj)| =











O n^−2γ+ν

if 1/2 < γ < 1, O n^−ca∗

if γ = 1 and 1/2 < ca^∗ < 2 − ν, O n^−2+νln(n)

if γ = 1 and ca^∗= 2 − ν, O n^−2+ν

if γ = 1 and ca^∗> 2 − ν .

Case ν > γ We have Gk+1,n−1(cαj, q) = O n^−(ν−γ)|F_k+1,n−1^ν (q)| + k^−(ν−γ)|F_k+1,n−1^γ (cαj)|
by means of LemmaA.4, and so we get

ρ²¹_n = X

j∈J, j6=1

1{2∈I_j}O n^−(ν−γ)

n−1

X

k=m₀

1

k^2γ|F_k+1,n−1^ν (q)| +

n−1

X

k=m₀

1

k^γ+ν|F_k+1,n−1^γ (cα_j)|

! ,

where, by LemmaA.2, the second term is O(n^−ν), while the sum in the first term has the asymptotic behavior given in (A.19).

Case ν = γ Assuming q 6= cαjfor all j ≥ 2, by LemmaA.4, we have²Gk+1,n−1(cαj, q) = O |F_k+1,n−1^γ (q)| +

|F_k+1,n−1^γ (cαj)|
, and so we get

ρ²¹_n = X

j∈J, j6=1

1{2∈Ij}O

n−1

X

k=m0

1

k^2γ|F_k+1,n−1^γ (q)| +

n−1

X

k=m0

1

k^2γ|F_k+1,n−1^γ (cαj)|

! ,

where, by LemmaA.2, we have for x = q or x ∈ {cαj: j ∈ J, j 6= 1}

n−1

X

k=m₀

1

k^2γ|F_k+1,n−1^γ (x)| =











O (n^−γ) if 1/2 < ν = γ < 1,

O (n^−ax) if ν = γ = 1 and 1/2 < ax< 1, O n⁻¹ln(n)

if ν = γ = 1 and ax= 1, O n⁻¹

if ν = γ = 1 and ax> 1 and so, setting x^∗:= min{q, ca^∗}, we can write

ρ²¹_n = X

j∈J, j6=1

1{2∈I_j}











O (n^−γ) if 1/2 < ν = γ < 1, O n^−x∗

if ν = γ = 1 and 1/2 < x^∗ < 1, O n⁻¹ln(n)

if ν = γ = 1 and x^∗= 1, O n⁻¹

if ν = γ = 1 and x^∗> 1 .

Summing up, taking into account the conditions ca^∗> 1/2 when γ = 1 and q > 1/2 when ν = 1, from the asymptotic behavior given above we easily obtain that in all the cases (i)-(v) we have t_n(J (I))

Pn−1

k=m₀ρ^{J (I)}_n,k   → 0 a.s. and so

t_n(J (I))

n−1

X

k=m0

ρ^{J (I)}_n,k −→ 0 .^a.s. (A.20)

In the case (vi), the evaluation of the asymptotic behavior given in (A.19) for the term ρ²²_n is not enough in order to conclude that tn(J (I))ρ²²_n → 0. Therefore, we need a better evaluation, that we can get applying Lemma A.2in a different way. Indeed, in the case (vi), taking u > 1 and applying Lemma A.2with e = u,

2If there exists j ≥ 2 such that q = cαj, we have to consider the other asymptotic expression given in LemmaA.4.

δ = ν and β = 2γu > 1, we find

t_n(J (I))ρ²²_n
^u

= n^uν/2O

n−1

X

k=m₀

k^−2γu|F_k+1,n−1^ν (q)|^u

!

= n^uν/2











O n^−2γu+ν

if 1/2 < ν < 1,

O (n^−qu) if ν = 1 and 1/2 < q < 2γ − u⁻¹, O (n^−quln(n)) if ν = 1 and q = 2γ − u⁻¹ > 1/2, O n^−2γu+1

if ν = 1 and q > max{1/2, 2γ − u⁻¹} .

Hence, from the above relations we get that it is possible to find u > 1 large enough such that (tn(J (I))ρ²²_n)^u→ 0, that trivially implies tn(J (I))ρ²²_n → 0. Therefore also in the case (vi), we can conclude that (A.20) holds true.

A.6. Computation of the limit d

Recall that we have d^j(1)_k,n =

(

rbk−1 for j = 1

rb_k−1F_k+1,n−1^γ (cα_j) for j ≥ 2 and

d^j(2)_k,n =

( (qbk,k−brk−1) F_k+1,n−1^ν (q) for j = 1

λ_jrb_k−1G_k+1,n−1(cα_j, q) + (qb_k,k−br_k−1g(λ_j)) F_k+1,n−1^ν (q) for j ≥ 2 , where, for each j ≥ 2, we have g(λ_j) = λ_j when ν < γ, while g(λ_j) = 0 when ν ≥ γ.

Here, for each of the six cases (i) − (vi) listed in the statement of (Aletti, Crimaldi and Ghiglietti,2019, Lemma 5.1), we compute the limit

d^{j1(i1),j2(i2)} = lim

n tn(J (I))²

n−1

X

k=m₀

d^j_k,n¹⁽ⁱ¹⁾d^j_k,n²⁽ⁱ²⁾.

For all the computations, we make the assumptions stated in (Aletti, Crimaldi and Ghiglietti,2019, Section 2) and we use LemmaA.3and LemmaA.4.

Case (i) Take ν < γ, j1, j2∈ {2, . . . , N } and i1= i2= 1. We have

limn tn(J (I))²

n−1

X

k=m0

d^j_k,n¹⁽ⁱ¹⁾d^j_k,n²⁽ⁱ²⁾= lim

n n^γ

n−1

X

k=m0

br²_k−1F_k+1,n−1^γ (cαj₁)F_k+1,n−1^γ (cαj₂)

= c²lim

n n^γ

n−1

X

k=m0

k^−2γF_k+1,n−1^γ (cα_j1)F_k+1,n−1^γ (cα_j2)

= c²

c(αj₁+ αj₂) −1{γ=1}

.

Case (ii) Take ν < γ, j1, j2∈ {1, . . . , N } and i1= i2= 2. For j1= j2= 1, we have

limn tn(J (I))²

n−1

X

k=m₀

d^j_k,n¹⁽ⁱ¹⁾d^j_k,n²⁽ⁱ²⁾= lim

n n^ν

n−1

X

k=m₀

(qbk,k−brk−1)²F_k+1,n−1^ν (q)²

= lim

n n^ν

n−1

X

k=m₀

bq_k,k² F_k+1,n−1^ν (q)²

= q²lim

n n^ν

n−1

X

k=m0

k^−2νF_k+1,n−1^ν (q)²= q 2.

(Note that the above second equality is due to the fact that some terms are o(n^−ν) and so we can cancel them.) Similarly, for the cases j1≥ 2, j2≥ 2 and j1= 1, j2≥ 2 and j1≥ 2, j2= 1, using LemmaA.4, which allows us to replace in the computation of the desired limit the quantity Gk+1,n−1(cαj, q) by

−cα_j q

F_k+1,n−1^ν (q)

(n − 1)^γ−ν −F_k+1,n−1^γ (cα_j) k^γ−ν

! , and removing the terms which are o(n^−ν), we obtain

lim

n t_n(J (I))²

n−1

X

k=m0

d^j_k,n¹⁽ⁱ¹⁾d^j_k,n²⁽ⁱ²⁾= lim

n n^ν

n−1

X

k=m0

bq_k,k² F_k+1,n−1^ν (q)²

= q²lim

n n^ν

n−1

X

k=m₀

k^−2νF_k+1,n−1^ν (q)²= q 2.

Case (iii) Take ν = γ, j1, j2 ∈ {1, . . . , N } and i1, i2 ∈ {1, 2} with ih 6= 1 if jh = 1. Recall that we are assuming q 6= cαj for all j ≥ 2³. Therefore, for j1= j2= 1 and i1= i2= 2, we have

limn t_n(J (I))²

n−1

X

k=m₀

d^j_k,n¹⁽ⁱ¹⁾d^j_k,n²⁽ⁱ²⁾= lim

n n^γ

n−1

X

k=m₀

(qb_k,k−rb_k−1)²F_k+1,n−1^γ (q)²

= lim

n (q − c)²n^γ

n−1

X

k=m₀

1

k^2γF_k+1,n−1^γ (q)²= (q − c)² 2q −1{γ=1}

.

For j1= 1, j2≥ 2, i1= 2 and i2= 1, we have limn tn(J (I))²

n−1

X

k=m₀

d^j_k,n¹⁽ⁱ¹⁾d^j_k,n²⁽ⁱ²⁾=

limn n^γ

n−1

X

k=m0

(qbk,k−brk−1) F_k+1,n−1^γ (q)rbk−1F_k+1,n−1^γ (cαj₂) =

(q − c)c lim

n n^γ

n−1

X

k=m0

1

k^2γF_k+1,n−1^γ (q)F_k+1,n−1^γ (cαj₂) = c(q − c) cαj₂+ q −1{γ=1}

.

By symmetry, for j₁≥ 2, j₂= 1, i₁= 1 and i₂= 2, we have d^{j1(i1),j2(i2)}= c(q − c)

cαj₁+ q −1{γ=1}

.

3In the case q = cαjfor some j ≥ 2 the computations are similar, but we have to consider the other asymptotic expression given in LemmaA.4.